Found problems: 85335
2003 China Team Selection Test, 3
Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that:
$\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$
2016 Azerbaijan JBMO TST, 1
If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality:
$$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$
2020 USMCA, 11
A permutation of $USMCAUSMCA$ is selected uniformly at random. What is the probability that this permutation is exactly one transposition away from $USMCAUSMCA$ (i.e. does not equal $USMCAUSMCA$, but can be turned into $USMCAUSMCA$ by swapping one pair of letters)?
2014 Purple Comet Problems, 3
The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy]
2009 Balkan MO Shortlist, A2
Let $ABCD$ be a square and points $M$ $\in$ $BC$, $N \in CD$, $P$ $\in$ $DA$, such that $\angle BAM$ $=$ $x$, $\angle CMN$ $=$ $2x$, $\angle DNP$ $=$ $3x$
[list=i]
[*] Show that, for any $x \in (0, \tfrac{\pi}{8} )$, such a configuration exists
[*] Determine the number of angles $x \in ( 0, \tfrac{\pi}{8} )$ for which $\angle APB =4x$
2004 Bulgaria Team Selection Test, 3
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
2002 Moldova National Olympiad, 2
For every nonnegative integer $ n$ and every real number $ x$ prove the inequality:
$ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
2002 Romania Team Selection Test, 1
Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$.
Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number.
[i]Laurentiu Panaitopol[/i]
2008 ITest, 69
In the sequence in the previous problem, how many of $u_1,u_2,u_3,\ldots, u_{2008}$ are pentagonal numbers?
2007 Harvard-MIT Mathematics Tournament, 3
Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$.
2009 Regional Olympiad of Mexico Center Zone, 3
An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of length $ 1$, drawing parallel lines (at distance $ 1$) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
Kyiv City MO Juniors 2003+ geometry, 2018.7.41
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
2004 AMC 10, 2
For any three real numbers $ a$, $ b$, and $ c$, with $ b\neq c$, the operation $ \otimes$ is defined by:
\[ \otimes(a,b,c) \equal{} \frac {a}{b \minus{} c}
\]What is $ \otimes(\otimes(1,2,3), \otimes(2,3,1),\otimes(3,1,2))$?
$ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(B)}\ \minus{}\!\frac {1}{4}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac {1}{4}\qquad
\textbf{(E)}\ \frac {1}{2}$
2023 Indonesia TST, 1
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2008 Peru IMO TST, 1
Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way.
Prove that $ La$ $ Lb$ and $ Lc$ are concurrent.
Daniel
1995 AIME Problems, 15
Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2014 Contests, 3
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.
2013 USA Team Selection Test, 4
Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.
2013 Princeton University Math Competition, 7
Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.
1987 Iran MO (2nd round), 2
Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that
\[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\]
Let $g$ be a function for which
\[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\]
Prove that $g$ is bounded.
2011 All-Russian Olympiad, 4
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]
2011 Saudi Arabia Pre-TST, 1.3
Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.
2005 Vietnam National Olympiad, 2
Find all triples of natural $ (x,y,n)$ satisfying the condition:
\[ \frac {x! \plus{} y!}{n!} \equal{} 3^n
\]
Define $ 0! \equal{} 1$
2022 Sharygin Geometry Olympiad, 10.1
$A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are two squares with their vertices arranged clockwise.The perpendicular bisector of segment $A_1B_1,A_2B_2,A_3B_3,A_4B_4$ and the perpendicular bisector of segment $A_2B_2,A_3B_3,A_4B_4,A_1B_1$ intersect at point $P,Q,R,S$ respectively.Show that:$PR\perp QS$.